If $\alpha, \beta$, where $\alpha < \beta$, are the roots of the equation $\lambda x^{2} - (\lambda + 3)x + 3 = 0$ such that $\frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}$, then the sum of all possible values of $\lambda$ is:

If $\alpha, \beta, \gamma$ are the roots of $x^3+ax^2+bx+c=0$,then find the value of $\sum \frac{1}{\alpha}$,given that $\alpha, \beta, \gamma$ are non-zero.

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$,with $\alpha>\beta$. For all positive integers $n$,define $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}, n \geq 1$ and $b_1=1$ and $b_n=a_{n-1}+a_{n+1}, n \geq 2$. Then which of the following options is/are correct?
$(1)$ $\sum_{i=1}^{n} a_i = a_{n+2}-1$ for all $n \geq 1$
$(2)$ $\sum_{n=1}^{\infty} \frac{a_n}{10^n} = \frac{10}{89}$
$(3)$ $\sum_{n=1}^{\infty} \frac{b_n}{10^n} = \frac{8}{89}$
$(4)$ $b_n = \alpha^n+\beta^n$ for all $n \geq 1$

If $p$ and $q$ are the roots of $x^2 + px + q = 0$,then

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-9x^2+23x-15=0$,then $\alpha^3+\beta^3+\gamma^3=$

If the ratio of the roots of the equations $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ are equal,then: